Concise Tensors of Minimal Border Rank

Abstract

We determine defining equations for the set of concise tensors of minimal border rank in C m⊗C m⊗C m when m = 5 and the set of concise minimal border rank 1∗-generic tensors when m = 5, 6. We solve the classical problem in algebraic complexity theory of classifying minimal border rank tensors in the special case m = 5. Our proofs utilize two recent developments: 111-equations defined by Buczyńska–Buczyński and results of Jelisiejew–Šivic on the variety ´ of commuting matrices. We introduce a new algebraic invariant of a concise tensor, its 111- algebra, and exploit it to give a strengthening of Friedland’s normal form for 1-degenerate tensors satisfying Strassen’s equations. We use the 111-algebra to characterize wild minimal border rank tensors and classify them in C 5 ⊗ C 5 ⊗ C 5.